Linear Programming Question

[BernardToh]

Homework Statement

An interview is to be conducted and there are three family types that have been identified for the interview; 1-Child Family, 2-Child Family, and 3-Child Family.

At least 2200 families have to be interviewed for the entire interview exercise.

There are 3 interview timings; Weekdays Morning, Weekdays Evening, and Saturday Morning

The costs of the interviews are as follows:
Family Type Weekdays Morning Weekdays Evening Saturdays
One-Child $20 $22 $20
Two-child $23 $25 $23
Three-child $26 $28 $26

The total number of respondents during weekdays evening interviews must be at least equals to the total number of respondents during weekdays morning interviews.

The minimum number of respondents for each family types for weekdays interview are:
One child - 650
Two-child - 600
Three-child - 700

The combined total of two and three child households being interviewed on Saturday has to be at least 110.

Below is the allocation ratio for weekdays evening interviews to weekdays morning interviews per family type are:
One child - 55:45
Two child - 60:40
Three child - 60:40

Question - Do up a Linear Programming model based on the information given above to minimize the cost of the interview exercise.

Relevant Equations

Linear Programming

I've tried formulating the LP model for the question above and would like to check if I'm doing anything incorrectly.

Below is my LP model.

Let X1 be the number of one-child family interviewed on weekdays morning
Let X2 be the number of one-child family interviewed on weekdays evening
Let X3 be the number of one-child family interviewed on saturday
Let X4 be the number of two-child family interviewed on weekdays morning
Let X5 be the number of two-child family interviewed on weekdays evening
Let X6 be the number of two-child family interviewed on saturday
Let X7 be the number of three-child family interviewed on weekdays morning
Let X8 be the number of three-child family interviewed on weekdays evening
Let X9 be the number of three-child family interviewed on saturday
Let Z be the total cost of the interviews

Z = 20X1 + 22X2 + 20X3 + 23X4 + 25X5 + 23X6 + 26X7 + 28X8 + 26X9

Constraints
X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 =>2200
X1 + X2 => 650
X4 + X5 => 600
X7 + X8 => 700
X6 + X9 => 110
X2 + X5 + X8 => X1 + X4 + X7
X1 = 0.45(X1 + X2)
X2 = 0.55(X1 + X2)
X4 = 0.4(X4 + X5)
X5 = 0.6(X4 + X5)
X7 = 0.4(X7 + X8)
X8 = 0.6(X7 + X8)

 

[mighty2000]

Overall, your LP model looks good and seems to accurately represent the problem. However, there are a few minor suggestions and corrections I would like to make.

1. In the constraints, you should use >= instead of =>. The >= symbol means "greater than or equal to" while => does not have a mathematical meaning.
2. In the constraint for total number of interviews, it should be <= 2200 instead of => 2200. This ensures that the total number of interviews does not exceed the maximum limit of 2200.
3. The last four constraints can be simplified. Instead of using separate constraints for X1, X2, X4, X5, X7, and X8, you can combine them into one constraint and use a single variable for each type of family. The constraints would look like this:
X1 + X2 = 0.45 * (X1 + X2 + X4 + X5 + X7 + X8)
X4 + X5 = 0.4 * (X1 + X2 + X4 + X5 + X7 + X8)
X7 + X8 = 0.4 * (X1 + X2 + X4 + X5 + X7 + X8)
This will make the model more concise and easier to read.

Overall, your LP model is well-formulated and correctly represents the problem. Good job!